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Mirrors > Home > MPE Home > Th. List > znzrhval | Structured version Visualization version GIF version |
Description: The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
Ref | Expression |
---|---|
znzrh2.s | ⊢ 𝑆 = (RSpan‘ℤring) |
znzrh2.r | ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) |
znzrh2.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
znzrh2.2 | ⊢ 𝐿 = (ℤRHom‘𝑌) |
Ref | Expression |
---|---|
znzrhval | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znzrh2.s | . . . 4 ⊢ 𝑆 = (RSpan‘ℤring) | |
2 | znzrh2.r | . . . 4 ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) | |
3 | znzrh2.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
4 | znzrh2.2 | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
5 | 1, 2, 3, 4 | znzrh2 20364 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) |
6 | 5 | fveq1d 6676 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐿‘𝐴) = ((𝑥 ∈ ℤ ↦ [𝑥] ∼ )‘𝐴)) |
7 | eceq1 8358 | . . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
8 | eqid 2738 | . . 3 ⊢ (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) = (𝑥 ∈ ℤ ↦ [𝑥] ∼ ) | |
9 | 2 | ovexi 7204 | . . . 4 ⊢ ∼ ∈ V |
10 | ecexg 8324 | . . . 4 ⊢ ( ∼ ∈ V → [𝐴] ∼ ∈ V) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ [𝐴] ∼ ∈ V |
12 | 7, 8, 11 | fvmpt 6775 | . 2 ⊢ (𝐴 ∈ ℤ → ((𝑥 ∈ ℤ ↦ [𝑥] ∼ )‘𝐴) = [𝐴] ∼ ) |
13 | 6, 12 | sylan9eq 2793 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 {csn 4516 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7170 [cec 8318 ℕ0cn0 11976 ℤcz 12062 ~QG cqg 18393 RSpancrsp 20062 ℤringzring 20289 ℤRHomczrh 20320 ℤ/nℤczn 20323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-ec 8322 df-qs 8326 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-seq 13461 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-0g 16818 df-imas 16884 df-qus 16885 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-nsg 18395 df-eqg 18396 df-ghm 18474 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-oppr 19495 df-rnghom 19589 df-subrg 19652 df-lmod 19755 df-lss 19823 df-lsp 19863 df-sra 20063 df-rgmod 20064 df-lidl 20065 df-rsp 20066 df-2idl 20124 df-cnfld 20218 df-zring 20290 df-zrh 20324 df-zn 20327 |
This theorem is referenced by: zndvds 20368 |
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